FixedEffect Versus RandomEffects Models


 Miles Patterson
 1 years ago
 Views:
Transcription
1 CHAPTER 13 FixedEffect Versus RandomEffects Models Introduction Definition of a summary effect Estimating the summary effect Extreme effect size in a large study or a small study Confidence interval The null hypothesis Which model should we use? Model should not be based on the test for heterogeneity Concluding remarks INTRODUCTION In Chapter 11 and Chapter 1 we introduced the fixedeffect and randomeffects models. Here, we highlight the conceptual and practical differences between them. Consider the forest plots in Figures 13.1 and 13.. They include the same six studies, but the first uses a fixedeffect analysis and the second a randomeffects analysis. These plots provide a context for the discussion that follows. DEFINITION OF A SUMMARY EFFECT Both plots show a summary effect on the bottom line, but the meaning of this summary effect is different in the two models. In the fixedeffect analysis we assumethatthetrueeffectsizeisthesame in all studies, and the summary effect is our estimate of this common effect size. In the randomeffects analysis we assume that the true effect size varies from one study to the next, and that the studies in our analysis represent a random sample of effect sizes that could Introduction to MetaAnalysis. Michael Borenstein, L. V. Hedges, J. P. T. Higgins and H. R. Rothstein 009 John Wiley & Sons, Ltd. ISBN:
2 78 FixedEffect Versus RandomEffects Models Impact of Intervention (Fixed effect) Std Diff (g) Relative Weight Standardized mean difference (g) and 95% confidence interval Carroll Grant Peck Donat Stewart Young Summary % 13% 8% 39% 10% 18% 100% Figure 13.1 Fixedeffect model forest plot showing relative weights. Impact of Intervention (Random effects) Std Diff (g) Relative Weight Standardized mean difference (g) with 95% confidence and prediction intervals Carroll Grant Peck Donat Stewart Young Summary % 16% 13% 3% 14% 18% 100% Figure 13. Randomeffects model forest plot showing relative weights. have been observed. The summary effect is our estimate of the mean of these effects. ESTIMATING THE SUMMARY EFFECT Under the fixedeffect model we assume that the true effect size for all studies is identical, and the only reason the effect size varies between studies is sampling error (error in estimating the effect size). Therefore, when assigning
3 Chapter 13: FixedEffect Versus RandomEffects Models weights to the different studies we can largely ignore the information in the smaller studies since we have better information about the same effect size in the larger studies. By contrast, under the randomeffects model the goal is not to estimate one true effect, but to estimate the mean of a distribution of effects. Since each study provides information about a different effect size, we want to be sure that all these effect sizes are represented in the summary estimate. This means that we cannot discount a small study by giving it a very small weight (the way we would in a fixedeffect analysis). The estimate provided by that study may be imprecise, but it is information about an effect that no other study has estimated. By the same logic we cannot give too much weight to a very large study (the way we might in a fixedeffect analysis). Our goal is to estimate the mean effect in a range of studies, and we do not want that overall estimate to be overly influenced by any one of them. In these graphs, the weight assigned to each study is reflected in the size of the box (specifically, the area) for that study. Under the fixedeffect model there is a wide range of weights (as reflected in the size of the boxes) whereas under the randomeffects model the weights fall in a relatively narrow range. For example, compare the weight assigned to the largest study (Donat) with that assigned to the smallest study (Peck) under the two models. Under the fixedeffect model Donat is given about five times as much weight as Peck. Under the randomeffects model Donat is given only 1.8 times as much weight as Peck. EXTREME EFFECT SIZE IN A LARGE STUDY OR A SMALL STUDY How will the selection of a model influence the overall effect size? In this example Donat is the largest study, and also happens to have the highest effect size. Under the fixedeffect model Donat was assigned a large share (39%) of the total weight and pulled the mean effect up to By contrast, under the randomeffects model Donat was assigned a relatively modest share of the weight (3%). It therefore had less pull on the mean, which was computed as Similarly, Carroll is one of the smaller studies and happens to have the smallest effect size. Under the fixedeffect model Carroll was assigned a relatively small proportion of the total weight (1%), and had little influence on the summary effect. By contrast, under the randomeffects model Carroll carried a somewhat higher proportion of the total weight (16%) and was able to pull the weighted mean toward the left. The operating premise, as illustrated in these examples, is that whenever is nonzero, the relative weights assigned under random effects will be more balanced than those assigned under fixed effects. As we move from fixed effect to random effects, extreme studies will lose influence if they are large, and will gain influence if they are small.
4 80 FixedEffect Versus RandomEffects Models CONFIDENCE INTERVAL Under the fixedeffect model the only source of uncertainty is the withinstudy (sampling or estimation) error. Under the randomeffects model there is this same source of uncertainty plus an additional source (betweenstudies variance). It follows that the variance, standard error, and confidence interval for the summary effect will always be larger (or wider) under the randomeffects model than under the fixedeffect model (unless T is zero, in which case the two models are the same). In this example, the standard error is for the fixedeffect model, and for the randomeffects model. Study A Study B Study C Study D Fixedeffect model Effect size and 95% confidence interval Summary Figure 13.3 Very large studies under fixedeffect model. Study A Study B Study C Study D Randomeffects model Effect size and 95% confidence interval Summary Figure 13.4 Very large studies under randomeffects model.
5 Chapter 13: FixedEffect Versus RandomEffects Models Consider what would happen if we had five studies, and each study had an infinitely large sample size. Under either model the confidence interval for the effect size in each study would have a width approaching zero, since we know the effect size in that study with perfect precision. Under the fixedeffect model the summary effect would also have a confidence interval with a width of zero, since we know the common effect precisely (Figure 13.3). By contrast, under the randomeffects model the width of the confidence interval would not approach zero (Figure 13.4). While we know the effect in each study precisely, these effects have been sampled from a universe of possible effect sizes, and provide only an estimate of the mean effect. Just as the error within a study will approach zero only as the sample size approaches infinity, so too the error of these studies as an estimate of the mean effect will approach zero only as the number of studies approaches infinity. More generally, it is instructive to consider what factors influence the standard error of the summary effect under the two models. The following formulas are based on a metaanalysis of means from k onegroup studies, but the conceptual argument applies to all metaanalyses. The withinstudy variance of each mean depends on the standard deviation (denoted ) of participants scores and the sample size of each study (n). For simplicity we assume that all of the studies have the same sample size and the same standard deviation (see Box 13.1 for details). Under the fixedeffect model the standard error of the summary effect is given by rffiffiffiffiffiffiffiffiffiffiffiffi SE M ¼ : ð13:1þ k n It follows that with a large enough sample size the standard error will approach zero, and this is true whether the sample size is concentrated on one or two studies, or dispersed across any number of studies. Under the randomeffects model the standard error of the summary effect is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SE M ¼ k n þ : ð13:þ k The first term is identical to that for the fixedeffect model and, again, with a large enough sample size, this term will approach zero. By contrast, the second term (which reflects the betweenstudies variance) will only approach zero as the number of studies approaches infinity. These formulas do not apply exactly in practice, but the conceptual argument does. Namely, increasing the sample size within studies is not sufficient to reduce the standard error beyond a certain point (where that point is determined by and k). If there is only a small number of studies, then the standard error could still be substantial even if the total n is in the tens of thousands or higher.
6 8 FixedEffect Versus RandomEffects Models BOX 13.1 FACTORS THAT INFLUENCE THE STANDARD ERROR OF THE SUMMARY EFFECT. To illustrate the concepts with some simple formulas, let us consider a metaanalysis of studies with the very simplest design, such that each study comprises a single sample of n observations with standard deviation. We combine estimates of the mean in a metaanalysis. The variance of each estimate is V Yi ¼ n so the (inversevariance) weight in a fixedeffect metaanalysis is W i ¼ 1 =n ¼ n and the variance of the summary effect under the fixedeffect model the standard error is given by V M ¼ X k 1 i¼1 W i ¼ 1 k n= ¼ k n : Therefore under the fixedeffect model the (true) standard error of the summary mean is given by SE M ¼ rffiffiffiffiffiffiffiffiffiffiffiffi : k n Under the randomeffects model the weight awarded to each study is W i ¼ 1 n þ and the (true) standard error of the summary mean turns out to be SE M rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k n þ : k
7 Chapter 13: FixedEffect Versus RandomEffects Models THE NULL HYPOTHESIS Often, after computing a summary effect, researchers perform a test of the null hypothesis. Under the fixedeffect model the null hypothesis being tested is that there is zero effect in every study. Under the randomeffects model the null hypothesis being tested is that the mean effect is zero. Although some may treat these hypotheses as interchangeable, they are in fact different, and it is imperative to choose the test that is appropriate to the inference a researcher wishes to make. WHICH MODEL SHOULD WE USE? The selection of a computational model should be based on our expectation about whether or not the studies share a common effect size and on our goals in performing the analysis. Fixed effect It makes sense to use the fixedeffect model if two conditions are met. First, we believe that all the studies included in the analysis are functionally identical. Second, our goal is to compute the common effect size for the identified population, and not to generalize to other populations. For example, suppose that a pharmaceutical company will use a thousand patients to compare a drug versus placebo. Because the staff can work with only 100 patients at a time, the company will run a series of ten trials with 100 patients in each. The studies are identical in the sense that any variables which can have an impact on the outcome are the same across the ten studies. Specifically, the studies draw patients from a common pool, using the same researchers, dose, measure, and so on (we assume that there is no concern about practice effects for the researchers, nor for the different starting times of the various cohorts). All the studies are expected to share a common effect and so the first condition is met. The goal of the analysis is to see if the drug works in the population from which the patients were drawn (and not to extrapolate to other populations), and so the second condition is met, as well. In this example the fixedeffect model is a plausible fit for the data and meets the goal of the researchers. It should be clear, however, that this situation is relatively rare. The vast majority of cases will more closely resemble those discussed immediately below. Random effects By contrast, when the researcher is accumulating data from a series of studies that had been performed by researchers operating independently, it would be unlikely that all the studies were functionally equivalent. Typically, the subjects or interventions in these studies would have differed in ways that would have impacted on
8 84 FixedEffect Versus RandomEffects Models the results, and therefore we should not assume a common effect size. Therefore, in these cases the randomeffects model is more easily justified than the fixedeffect model. Additionally, the goal of this analysis is usually to generalize to a range of scenarios. Therefore, if one did make the argument that all the studies used an identical, narrowly defined population, then it would not be possible to extrapolate from this population to others, and the utility of the analysis would be severely limited. A caveat There is one caveat to the above. If the number of studies is very small, then the estimate of the betweenstudies variance ( ) will have poor precision. While the randomeffects model is still the appropriate model, we lack the information needed to apply it correctly. In this case the reviewer may choose among several options, each of them problematic. One option is to report the separate effects and not report a summary effect. The hope is that the reader will understand that we cannot draw conclusions about the effect size and its confidence interval. The problem is that some readers will revert to vote counting (see Chapter 8) and possibly reach an erroneous conclusion. Another option is to perform a fixedeffect analysis. This approach would yield a descriptive analysis of the included studies, but would not allow us to make inferences about a wider population. The problem with this approach is that (a) we do want to make inferences about a wider population and (b) readers will make these inferences even if they are not warranted. A third option is to take a Bayesian approach, where the estimate of is based on data from outside of the current set of studies. This is probably the best option, but the problem is that relatively few researchers have expertise in Bayesian metaanalysis. Additionally, some researchers have a philosophical objection to this approach. For a more general discussion of this issue see When does it make sense to perform a metaanalysis in Chapter 40. MODEL SHOULD NOT BE BASED ON THE TEST FOR HETEROGENEITY In the next chapter we will introduce a test of the null hypothesis that the betweenstudies variance is zero. This test is based on the amount of betweenstudies variance observed, relative to the amount we would expect if the studies actually shared a common effect size. Some have adopted the practice of starting with a fixedeffect model and then switching to a randomeffects model if the test of homogeneity is statistically significant. This practice should be strongly discouraged because the decision to use the randomeffects model should be based on our understanding of whether or not all studies share a common effect size, and not on the outcome of a statistical test (especially since the test for heterogeneity often suffers from low power).
9 Chapter 13: FixedEffect Versus RandomEffects Models If the study effect sizes are seen as having been sampled from a distribution of effect sizes, then the randomeffects model, which reflects this idea, is the logical one to use. If the betweenstudies variance is substantial (and statistically significant) then the fixedeffect model is inappropriate. However, even if the betweenstudies variance does not meet the criterion for statistical significance (which may be due simply to low power) we should still take account of this variance when assigning weights. If T turns out to be zero, then the randomeffects analysis reduces to the fixedeffect analysis, and so there is no cost to using this model. On the other hand, if one has elected to use the fixedeffect model a priori but the test of homogeneity is statistically significant, then it would be important to revisit the assumptions that led to the selection of a fixedeffect model. CONCLUDING REMARKS Our discussion of differences between the fixedmodel and the randomeffects model focused largely on the computation of a summary effect and the confidence intervals for the summary effect. We did not address the implications of the dispersion itself. Under the fixedeffect model we assume that all dispersion in observed effects is due to sampling error, but under the randomeffects model we allow that some of that dispersion reflects real differences in effect size across studies. In the chapters that follow we discuss methods to quantify that dispersion and to consider its substantive implications. Although throughout this book we define a fixedeffect metaanalysis as assuming that every study has a common true effect size, some have argued that the fixedeffect method is valid without making this assumption. The point estimate of the effect in a fixedeffect metaanalysis is simply a weighted average and does not strictly require the assumption that all studies estimate the same thing. For simplicity and clarity we adopt a definition of a fixedeffect metaanalysis that does assume homogeneity of effect. SUMMARY POINTS A fixedeffect metaanalysis estimates a single effect that is assumed to be common to every study, while a randomeffects metaanalysis estimates the mean of a distribution of effects. Study weights are more balanced under the randomeffects model than under the fixedeffect model. Large studies are assigned less relative weight and small studies are assigned more relative weight as compared with the fixedeffect model. The standard error of the summary effect and (it follows) the confidence intervals for the summary effect are wider under the randomeffects model than under the fixedeffect model.
10 86 FixedEffect Versus RandomEffects Models The selection of a model must be based solely on the question of which model fits the distribution of effect sizes, and takes account of the relevant source(s) of error. When studies are gathered from the published literature, the randomeffects model is generally a more plausible match. The strategy of starting with a fixedeffect model and then moving to a randomeffects model if the test for heterogeneity is significant is a mistake, and should be strongly discouraged.
Systematic Reviews and Metaanalyses
Systematic Reviews and Metaanalyses Introduction A systematic review (also called an overview) attempts to summarize the scientific evidence related to treatment, causation, diagnosis, or prognosis of
More informationConfidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
More informationConfidence Intervals for the Difference Between Two Means
Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means
More informationCONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
More informationPooling and Metaanalysis. Tony O Hagan
Pooling and Metaanalysis Tony O Hagan Pooling Synthesising prior information from several experts 2 Multiple experts The case of multiple experts is important When elicitation is used to provide expert
More informationWISE Power Tutorial All Exercises
ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II
More informationA POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
More informationUCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates
UCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally
More information2 Precisionbased sample size calculations
Statistics: An introduction to sample size calculations Rosie Cornish. 2006. 1 Introduction One crucial aspect of study design is deciding how big your sample should be. If you increase your sample size
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationEffect Size Calculations and Elementary MetaAnalysis
Effect Size Calculations and Elementary MetaAnalysis David B. Wilson, PhD George Mason University August 2011 The EndGame ForestPlot of OddsRatios and 95% Confidence Intervals for the Effects of CognitiveBehavioral
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More information1. How different is the t distribution from the normal?
Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. tdistributions.
More informationMINITAB ASSISTANT WHITE PAPER
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationHypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the
More informationTwosample ttests.  Independent samples  Pooled standard devation  The equal variance assumption
Twosample ttests.  Independent samples  Pooled standard devation  The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular
More informationSTATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI
STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members
More informationMethods for Metaanalysis in Medical Research
Methods for Metaanalysis in Medical Research Alex J. Sutton University of Leicester, UK Keith R. Abrams University of Leicester, UK David R. Jones University of Leicester, UK Trevor A. Sheldon University
More informationWeb appendix: Supplementary material. Appendix 1 (online): Medline search strategy
Web appendix: Supplementary material Appendix 1 (online): Medline search strategy exp Venous Thrombosis/ Deep vein thrombosis.mp. Pulmonary embolism.mp. or exp Pulmonary Embolism/ recurrent venous thromboembolism.mp.
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationUNDERSTANDING THE INDEPENDENTSAMPLES t TEST
UNDERSTANDING The independentsamples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in reallife applications that they have been given their own names.
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationConfidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationSTATISTICAL ANALYSIS AND INTERPRETATION OF DATA COMMONLY USED IN EMPLOYMENT LAW LITIGATION
STATISTICAL ANALYSIS AND INTERPRETATION OF DATA COMMONLY USED IN EMPLOYMENT LAW LITIGATION C. Paul Wazzan Kenneth D. Sulzer ABSTRACT In employment law litigation, statistical analysis of data from surveys,
More informationExtending Hypothesis Testing. pvalues & confidence intervals
Extending Hypothesis Testing pvalues & confidence intervals So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationPower and Sample Size Determination
Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,
More informationThomas Stratmann Mercatus Center Scholar Professor of Economics, George Mason University
Thomas Stratmann Mercatus Center Scholar Professor of Economics, George Mason University to Representative Scott Garrett Chairman of the Capital Markets and Government Sponsored Enterprises Subcommittee
More informationSocial Studies 201 Notes for November 19, 2003
1 Social Studies 201 Notes for November 19, 2003 Determining sample size for estimation of a population proportion Section 8.6.2, p. 541. As indicated in the notes for November 17, when sample size is
More informationConstructing and Interpreting Confidence Intervals
Constructing and Interpreting Confidence Intervals Confidence Intervals In this power point, you will learn: Why confidence intervals are important in evaluation research How to interpret a confidence
More informationConfidence Intervals for Cpk
Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process
More informationSoftware for Publication Bias
CHAPTER 11 Software for Publication Bias Michael Borenstein Biostat, Inc., USA KEY POINTS Various procedures for addressing publication bias are discussed elsewhere in this volume. The goal of this chapter
More informationSample size estimation is an important concern
Sample size and power calculations made simple Evie McCrumGardner Background: Sample size estimation is an important concern for researchers as guidelines must be adhered to for ethics committees, grant
More informationHypothesis testing for µ:
University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
More information12: Analysis of Variance. Introduction
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
More informationSample Size and Power in Clinical Trials
Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance
More informationAP Statistics 2001 Solutions and Scoring Guidelines
AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationPlanning sample size for randomized evaluations
TRANSLATING RESEARCH INTO ACTION Planning sample size for randomized evaluations Simone Schaner Dartmouth College povertyactionlab.org 1 Course Overview Why evaluate? What is evaluation? Outcomes, indicators
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More informationConfidence Interval: pˆ = E = Indicated decision: < p <
Hypothesis (Significance) Tests About a Proportion Example 1 The standard treatment for a disease works in 0.675 of all patients. A new treatment is proposed. Is it better? (The scientists who created
More informationOnline 12  Sections 9.1 and 9.2Doug Ensley
Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics  Ensley Assignment: Online 12  Sections 9.1 and 9.2 1. Does a Pvalue of 0.001 give strong evidence or not especially strong
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationPUBLIC HEALTH OPTOMETRY ECONOMICS. Kevin D. Frick, PhD
Chapter Overview PUBLIC HEALTH OPTOMETRY ECONOMICS Kevin D. Frick, PhD This chapter on public health optometry economics describes the positive and normative uses of economic science. The terms positive
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationGeneral Method: Difference of Means. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n 1, n 2 ) 1.
General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n
More informationChapter 7 Section 7.1: Inference for the Mean of a Population
Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used
More informationHow MetaAnalysis Increases Statistical Power
Psychological Methods Copyright 2003 by the American Psychological Association, Inc. 2003, Vol. 8, No. 3, 243 253 1082989X/03/$12.00 DOI: 10.1037/1082989X.8.3.243 How MetaAnalysis Increases Statistical
More informationMultivariate Analysis of Ecological Data
Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology
More information9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d
CHAPTER 9 Comparison of Paired Samples 9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d =.68 4 =.34. (b) H 0 : The mean
More informationELEMENTARY STATISTICS IN CRIMINAL JUSTICE RESEARCH: THE ESSENTIALS
ELEMENTARY STATISTICS IN CRIMINAL JUSTICE RESEARCH: THE ESSENTIALS 2005 James A. Fox Jack Levin ISBN 0205420532 (Please use above number to order your exam copy.) Visit www.ablongman.com/replocator
More informationGlossary of Terms Ability Accommodation Adjusted validity/reliability coefficient Alternate forms Analysis of work Assessment Battery Bias
Glossary of Terms Ability A defined domain of cognitive, perceptual, psychomotor, or physical functioning. Accommodation A change in the content, format, and/or administration of a selection procedure
More information11. Logic of Hypothesis Testing
11. Logic of Hypothesis Testing A. Introduction B. Significance Testing C. Type I and Type II Errors D. One and TwoTailed Tests E. Interpreting Significant Results F. Interpreting NonSignificant Results
More informationT O P I C 1 1 Introduction to statistics Preview Introduction In previous topics we have looked at ways of gathering data for research purposes and ways of organising and presenting it. In topics 11 and
More informationPASS Sample Size Software
Chapter 250 Introduction The Chisquare test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationHypothesis Testing. Learning Objectives. After completing this module, the student will be able to
Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationAnalysis Issues II. Mary Foulkes, PhD Johns Hopkins University
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationApixaban for the prevention of stroke and systemic embolism in people with nonvalvular atrial fibrillation
Apixaban for the prevention of stroke and systemic embolism in people with nonvalvular atrial fibrillation ERRATUM This report was commissioned by the NIHR HTA Programme as project number 11/49 This document
More information9. Sampling Distributions
9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationComparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the pvalue and a posterior
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationAP Statistics 2002 Scoring Guidelines
AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought
More informationSTATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4
STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate
More informationReview of basic statistics and the simplest forecasting model: the sample mean
Review of basic statistics and the simplest forecasting model: the sample mean Robert Nau Fuqua School of Business, Duke University August 2014 Most of what you need to remember about basic statistics
More informationMeasurement in ediscovery
Measurement in ediscovery A Technical White Paper Herbert Roitblat, Ph.D. CTO, Chief Scientist Measurement in ediscovery From an informationscience perspective, ediscovery is about separating the responsive
More informationThreeStage Phase II Clinical Trials
Chapter 130 ThreeStage Phase II Clinical Trials Introduction Phase II clinical trials determine whether a drug or regimen has sufficient activity against disease to warrant more extensive study and development.
More informationHow to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing ShortTerm Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
More informationStatistical Inference and ttests
1 Statistical Inference and ttests Objectives Evaluate the difference between a sample mean and a target value using a onesample ttest. Evaluate the difference between a sample mean and a target value
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
More informationTrial Sequential Analysis (TSA)
User manual for Trial Sequential Analysis (TSA) Kristian Thorlund, Janus Engstrøm, Jørn Wetterslev, Jesper Brok, Georgina Imberger, and Christian Gluud Copenhagen Trial Unit Centre for Clinical Intervention
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationTutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrclmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
More informationA Bayesian hierarchical surrogate outcome model for multiple sclerosis
A Bayesian hierarchical surrogate outcome model for multiple sclerosis 3 rd Annual ASA New Jersey Chapter / Bayer Statistics Workshop David Ohlssen (Novartis), Luca Pozzi and Heinz Schmidli (Novartis)
More informationWhat are confidence intervals and pvalues?
What is...? series Second edition Statistics Supported by sanofiaventis What are confidence intervals and pvalues? Huw TO Davies PhD Professor of Health Care Policy and Management, University of St Andrews
More informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More informationBinary Diagnostic Tests Two Independent Samples
Chapter 537 Binary Diagnostic Tests Two Independent Samples Introduction An important task in diagnostic medicine is to measure the accuracy of two diagnostic tests. This can be done by comparing summary
More informationName: Date: Use the following to answer questions 34:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationTechnology StepbyStep Using StatCrunch
Technology StepbyStep Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate
More informationA Review of Cross Sectional Regression for Financial Data You should already know this material from previous study
A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study But I will offer a review, with a focus on issues which arise in finance 1 TYPES OF FINANCIAL
More informationAP Statistics 2011 Scoring Guidelines
AP Statistics 2011 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in
More informationINTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the oneway ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
More informationRandomized Block Analysis of Variance
Chapter 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of
More informationBUS/ST 350 Exam 3 Spring 2012
BUS/ST 350 Exam 3 Spring 2012 Name Lab Section ID # INSTRUCTIONS: Write your name, lab section #, and ID# above. Note the statement at the bottom of this page that you must sign when you are finished with
More informationExperimental Designs (revisited)
Introduction to ANOVA Copyright 2000, 2011, J. Toby Mordkoff Probably, the best way to start thinking about ANOVA is in terms of factors with levels. (I say this because this is how they are described
More informationIntroduction to Hypothesis Testing OPRE 6301
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
More informationBA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420
BA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420 1. Which of the following will increase the value of the power in a statistical test
More information